Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sample-path large deviations for a class of heavy-tailed Markov additive processes

Bohan Chen, Chang-Han Rhee, Bert Zwart

Abstract

For a class of additive processes driven by the affine recursion $X_{n+1} = A_n X_n + B_n$, we develop a sample-path large deviations principle in the $M_1'$ topology on $D [0,1]$. We allow $B_n$ to have both signs and focus on the case where Kesten's condition holds on $A_1$, leading to heavy-tailed distributions. The most likely paths in our large deviations results are step functions with both positive and negative jumps.

Sample-path large deviations for a class of heavy-tailed Markov additive processes

Abstract

For a class of additive processes driven by the affine recursion , we develop a sample-path large deviations principle in the topology on . We allow to have both signs and focus on the case where Kesten's condition holds on , leading to heavy-tailed distributions. The most likely paths in our large deviations results are step functions with both positive and negative jumps.

Paper Structure

This paper contains 14 sections, 28 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background from Markov chain theory
  4. A useful change of measure
  5. M-convergence
  6. Main results
  7. Tail estimates on the area under the first return time/regeneration cycle
  8. One-sided large deviations
  9. Two-sided large deviations
  10. Proofs of Section \ref{['SECTION: PRELIMINARIES']}
  11. Proofs of Section \ref{['SECTION: TAIL ESTIMATES FOR THE AREA UNDER FIRST RETURN TIME/REGENERATION CYCYLE']}
  12. Proofs of Sections \ref{['SECTION: ONE-SIDED LARGE DEVIATIONS']} and \ref{['SECTION: TWO-SIDED LARGE DEVIATIONS']}
  13. Proof of Theorem \ref{['thm: one-sided large deviations']}
  14. Proof of Theorem \ref{['thm:two-sided-large-deviations']}

Key Result

Proposition 2.5

Assume that one of the following conditions hold. Then, for any $x_0 \in \mathds{R}$, there exists $\epsilon = \epsilon(x_0)$, $\theta>0$, and an open interval $E_0$ such that

Theorems & Definitions (60)

  • Remark 2.2
  • Proposition 2.5
  • Lemma 2.6
  • Remark 2.8
  • Lemma 2.9
  • Definition 2.11
  • Remark 2.12
  • Lemma 2.13
  • Theorem 3.1
  • Theorem 3.2
  • ...and 50 more